X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=07a70b0a2b651a40c207d5d3bcdfd807d55f92df;hb=0b6e486f52b6c42f78ba408543be0cc4b66fada7;hp=ff3ec0001b4c3508ceef2c43afb6d13ffb7d403f;hpb=428ef4a28dc25409df02f6af024043c21307a646;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index ff3ec00..07a70b0 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,15 +4,15 @@ Symmetric linear games and their solutions. This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ - from cvxopt import matrix, printing, solvers from .cones import CartesianProduct from .errors import GameUnsolvableException, PoorScalingException from .matrices import (append_col, append_row, condition_number, identity, - inner_product) -from . import options + inner_product, norm, specnorm) +from .options import ABS_TOL, FLOAT_FORMAT, DEBUG_FLOAT_FORMAT + +printing.options['dformat'] = FLOAT_FORMAT -printing.options['dformat'] = options.FLOAT_FORMAT class Solution: """ @@ -23,7 +23,7 @@ class Solution: -------- >>> print(Solution(10, matrix([1,2]), matrix([3,4]))) - Game value: 10.0000000 + Game value: 10.000... Player 1 optimal: [ 1] [ 2] @@ -323,6 +323,8 @@ class SymmetricLinearGame: if not self._e2 in K: raise ValueError('the point e2 must lie in the interior of K') + # Initial value of cached method. + self._L_specnorm_value = None def __str__(self): @@ -335,12 +337,12 @@ class SymmetricLinearGame: ' e1 = {:s},\n' \ ' e2 = {:s},\n' \ ' Condition((L, K, e1, e2)) = {:f}.' - indented_L = '\n '.join(str(self._L).splitlines()) - indented_e1 = '\n '.join(str(self._e1).splitlines()) - indented_e2 = '\n '.join(str(self._e2).splitlines()) + indented_L = '\n '.join(str(self.L()).splitlines()) + indented_e1 = '\n '.join(str(self.e1()).splitlines()) + indented_e2 = '\n '.join(str(self.e2()).splitlines()) return tpl.format(indented_L, - str(self._K), + str(self.K()), indented_e1, indented_e2, self.condition()) @@ -581,7 +583,7 @@ class SymmetricLinearGame: return matrix(0, (self.dimension(), 1), tc='d') - def _A(self): + def A(self): """ Return the matrix ``A`` used in our CVXOPT construction. @@ -609,12 +611,12 @@ class SymmetricLinearGame: >>> e1 = [1,1,1] >>> e2 = [1,2,3] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._A()) + >>> print(SLG.A()) [0.0000000 1.0000000 2.0000000 3.0000000] """ - return matrix([0, self._e2], (1, self.dimension() + 1), 'd') + return matrix([0, self.e2()], (1, self.dimension() + 1), 'd') @@ -657,7 +659,7 @@ class SymmetricLinearGame: """ identity_matrix = identity(self.dimension()) return append_row(append_col(self._zero(), -identity_matrix), - append_col(self._e1, -self._L)) + append_col(self.e1(), -self.L())) def _c(self): @@ -698,7 +700,7 @@ class SymmetricLinearGame: return matrix([-1, self._zero()]) - def _C(self): + def C(self): """ Return the cone ``C`` used in our CVXOPT construction. @@ -720,7 +722,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._C()) + >>> print(SLG.C()) Cartesian product of dimension 6 with 2 factors: * Nonnegative orthant in the real 3-space * Nonnegative orthant in the real 3-space @@ -770,7 +772,7 @@ class SymmetricLinearGame: @staticmethod - def _b(): + def b(): """ Return the ``b`` vector used in our CVXOPT construction. @@ -801,7 +803,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._b()) + >>> print(SLG.b()) [1.0000000] @@ -809,6 +811,57 @@ class SymmetricLinearGame: return matrix([1], tc='d') + def player1_start(self): + """ + Return a feasible starting point for player one. + + This starting point is for the CVXOPT formulation and not for + the original game. The basic premise is that if you normalize + :meth:`e2`, then you get a point in :meth:`K` that makes a unit + inner product with :meth:`e2`. We then get to choose the primal + objective function value such that the constraint involving + :meth:`L` is satisfied. + """ + p = self.e2() / (norm(self.e2()) ** 2) + dist = self.K().ball_radius(self.e1()) + nu = - self._L_specnorm()/(dist*norm(self.e2())) + x = matrix([nu, p], (self.dimension() + 1, 1)) + s = - self._G()*x + + return {'x': x, 's': s} + + + def player2_start(self): + """ + Return a feasible starting point for player two. + """ + q = self.e1() / (norm(self.e1()) ** 2) + dist = self.K().ball_radius(self.e2()) + omega = self._L_specnorm()/(dist*norm(self.e1())) + y = matrix([omega]) + z2 = q + z1 = y*self.e2() - self.L().trans()*z2 + z = matrix([z1, z2], (self.dimension()*2, 1)) + + return {'y': y, 'z': z} + + + def _L_specnorm(self): + """ + Compute the spectral norm of ``L`` and cache it. + """ + if self._L_specnorm_value is None: + self._L_specnorm_value = specnorm(self.L()) + return self._L_specnorm_value + + def tolerance_scale(self, solution): + # Don't return anything smaller than 1... we can't go below + # out "minimum tolerance." + norm_p1_opt = norm(solution.player1_optimal()) + norm_p2_opt = norm(solution.player2_optimal()) + scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt) + return max(1, scale/2.0) + def solution(self): """ @@ -844,11 +897,11 @@ class SymmetricLinearGame: >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: -6.1724138 + Game value: -6.172... Player 1 optimal: - [ 0.551...] - [-0.000...] - [ 0.448...] + [0.551...] + [0.000...] + [0.448...] Player 2 optimal: [0.448...] [0.000...] @@ -864,7 +917,7 @@ class SymmetricLinearGame: >>> e2 = [4,5,6] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: 0.0312500 + Game value: 0.031... Player 1 optimal: [0.031...] [0.062...] @@ -900,8 +953,8 @@ class SymmetricLinearGame: >>> print(SLG.solution()) Game value: 18.767... Player 1 optimal: - [-0.000...] - [ 9.766...] + [0.000...] + [9.766...] Player 2 optimal: [1.047...] [0.000...] @@ -918,28 +971,68 @@ class SymmetricLinearGame: >>> print(SLG.solution()) Game value: 24.614... Player 1 optimal: - [ 6.371...] - [-0.000...] + [6.371...] + [0.000...] Player 2 optimal: [2.506...] [0.000...] + This is another one that was difficult numerically, and caused + trouble even after we fixed the first two:: + + >>> from dunshire import * + >>> L = [[57.22233908627052301199, 41.70631373437460354126], + ... [83.04512571985074487202, 57.82581810406928468637]] + >>> K = NonnegativeOrthant(2) + >>> e1 = [7.31887017043399268346, 0.89744171905822367474] + >>> e2 = [0.11099824781179848388, 6.12564670639315345113] + >>> SLG = SymmetricLinearGame(L,K,e1,e2) + >>> print(SLG.solution()) + Game value: 70.437... + Player 1 optimal: + [9.009...] + [0.000...] + Player 2 optimal: + [0.136...] + [0.000...] + + And finally, here's one that returns an "optimal" solution, but + whose primal/dual objective function values are far apart:: + + >>> from dunshire import * + >>> L = [[ 6.49260076597376212248, -0.60528030227678542019], + ... [ 2.59896077096751731972, -0.97685530240286766457]] + >>> K = IceCream(2) + >>> e1 = [1, 0.43749513972645248661] + >>> e2 = [1, 0.46008379832200291260] + >>> SLG = SymmetricLinearGame(L, K, e1, e2) + >>> print(SLG.solution()) + Game value: 11.596... + Player 1 optimal: + [ 1.852...] + [-1.852...] + Player 2 optimal: + [ 1.777...] + [-1.777...] + """ try: - opts = {'show_progress': options.VERBOSE} + opts = {'show_progress': False} soln_dict = solvers.conelp(self._c(), self._G(), self._h(), - self._C().cvxopt_dims(), - self._A(), - self._b(), + self.C().cvxopt_dims(), + self.A(), + self.b(), + primalstart=self.player1_start(), + dualstart=self.player2_start(), options=opts) except ValueError as error: if str(error) == 'math domain error': # Oops, CVXOPT tried to take the square root of a # negative number. Report some details about the game # rather than just the underlying CVXOPT crash. - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise PoorScalingException(self) else: raise error @@ -964,9 +1057,23 @@ class SymmetricLinearGame: # that CVXOPT is convinced the problem is infeasible (and that # cannot happen). if soln_dict['status'] in ['primal infeasible', 'dual infeasible']: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) + # For the game value, we could use any of: + # + # * p1_value + # * p2_value + # * (p1_value + p2_value)/2 + # * the game payoff + # + # We want the game value to be the payoff, however, so it + # makes the most sense to just use that, even if it means we + # can't test the fact that p1_value/p2_value are close to the + # payoff. + payoff = self.payoff(p1_optimal, p2_optimal) + soln = Solution(payoff, p1_optimal, p2_optimal) + # The "optimal" and "unknown" results, we actually treat the # same. Even if CVXOPT bails out due to numerical difficulty, # it will have some candidate points in mind. If those @@ -977,29 +1084,18 @@ class SymmetricLinearGame: # close enough (one could be low by ABS_TOL, the other high by # it) because otherwise CVXOPT might return "unknown" and give # us two points in the cone that are nowhere near optimal. - if abs(p1_value - p2_value) > 2*options.ABS_TOL: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + # + if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL: + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) # And we also check that the points it gave us belong to the # cone, just in case... if (p1_optimal not in self._K) or (p2_optimal not in self._K): - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) - # For the game value, we could use any of: - # - # * p1_value - # * p2_value - # * (p1_value + p2_value)/2 - # * the game payoff - # - # We want the game value to be the payoff, however, so it - # makes the most sense to just use that, even if it means we - # can't test the fact that p1_value/p2_value are close to the - # payoff. - payoff = self.payoff(p1_optimal, p2_optimal) - return Solution(payoff, p1_optimal, p2_optimal) + return soln def condition(self): @@ -1030,13 +1126,11 @@ class SymmetricLinearGame: >>> e1 = [1] >>> e2 = e1 >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> actual = SLG.condition() - >>> expected = 1.8090169943749477 - >>> abs(actual - expected) < options.ABS_TOL - True + >>> SLG.condition() + 1.809... """ - return (condition_number(self._G()) + condition_number(self._A()))/2 + return (condition_number(self._G()) + condition_number(self.A()))/2 def dual(self): @@ -1072,10 +1166,10 @@ class SymmetricLinearGame: Condition((L, K, e1, e2)) = 44.476... """ - # We pass ``self._L`` right back into the constructor, because + # We pass ``self.L()`` right back into the constructor, because # it will be transposed there. And keep in mind that ``self._K`` # is its own dual. - return SymmetricLinearGame(self._L, - self._K, - self._e2, - self._e1) + return SymmetricLinearGame(self.L(), + self.K(), + self.e2(), + self.e1())